Since Rota formalised the theory [1]}, [2]} (on the grounds of previous contributions [3]}, [4]})
the standard framework for the theory has been that of posets, but the theory has also been
used in the context of monoids [5]}, and in the more
general framework of certain categories called Möbius categories,
introduced by Leroux [6]}. The uniform appearance of
the Möbius inversion formula across all application areas led
Lawvere [7]} in the 1980s to discover that
there is a universal Möbius function which induces all other
Möbius functions. It is an `arithmetic function' on a certain Hopf
algebra of Möbius intervals. A category is an interval if it has
an initial and a terminal object and the Möbius condition is a certain finiteness condition [8]}. This Hopf algebra has the property
that it receives a canonical coalgebra homomorphism from every
incidence coalgebra of a Möbius category. This includes all
locally finite posets and all the monoids considered in
[5]}. Lawvere's work remained unpublished for some
decades, but it is cited in influential texts from that time, such
as Joyal [10]} and Joni–Rota [2]}.
Independently, Ehrenborg [12]} constructed a closely
related Hopf algebra, but less universal. It only accounts for
intervals in posets. In both cases, the universal object can be
interpreted as the colimit of all incidence coalgebras of intervals.
The possibility of this is closely related to the local nature of
coalgebras, expressed for example in the well-known fact that every
coalgebra is the colimit of its finite-dimensional subcoalgebras,
see Sweedler [13]}.
